The main lakes in the Tetons are right along the road and Wyoming's nonresident fishing license fees are expensive. I'd plan to fish in backcountry lakes and streams in Yellowstone, where the Wyoming license is not needed, and which are more likely to be snow-free. There are only a few lakes in the Teton backcountry and they will still be frozen (and snowed in) in early June.
Do check on season opening dates, though; your early June time frame mentioned in your other post might be too early. Check out the Yellowstone NP website on nps.gov.
May your trails be crooked, winding, lonesome, dangerous, leading to the most amazing view--E. Abbey